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## Rostami2017 AOM Moist Convection RSWmodel.pdf Page 1 2 3 45629

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November 18, 2016

Geophysical and Astrophysical Fluid Dynamics
http://dx.doi.org/10.1080/03091929.2016.1269897

oneandtwolayer˙revision4

instabilities of vortices in the mcrsw model

2.

3

The model and the background flow

In this section we present the model we are working with. It is a rotating shallow water model
where convective fluxes due to the latent heat release are added at the interface between the
lower, humid, and the upper, dry layer, as well as surface evaporation at the lower boundary.
Although multi-layer generalisation of the model is possible, cf. the references in sect. 1 above,
we will be using the two- layer version, and its one-layer reduction. The model retains the
most rough features of the moist convection, yet in a self-consistent way, and allows to follow
both vortex and inertia-gravity wave components of the flow.

2.1.

Moist-convective rotating shallow water model

2.1.1. Equations of motion
The equations of the two-layer mcRSW, as introduced in (Lambaerts et al. 2011a) are:

D1 v 1

+ f zˆ × v 1 = −g∇H (h1 + h2 ),

Dt

v1 − v2
D2 v 2

+ f zˆ × v 2 = −g∇H (h1 + sh2 ) +
βP1 ,

h2
 Dt

∂t h1 + ∇.(h1 v 1 ) = −βP1 ,

∂t h2 + ∇.(h2 v 2 ) = +βP1 ,

∂t Q1 + ∇.(Q1 v 1 ) = −P1 + E.

(1)

Here v i is the horizontal velocity field in layer i = 1, 2 (counted from the bottom) and
Di /Dt is the corresponding horizontal material derivative. f is the Coriolis parameter, which
will be supposed to be constant, zˆ is the unit vector in z-direction, hi are thicknesses of
the layers, θi is the normalised potential temperature in each layer, s = (θ2 /θ1 ) &gt; 1 is the
stratification parameter. Let us recall that this model is obtained from the primitive equations,
with pseudo-height as vertical coordinate, by vertical averaging between the material surfaces
zi−1 , zi , with zi − zi−1 = hi and adding additional convective flux due to the latent heat
release through z1 . This convective flux is then linked to the water-vapour condensation P1
in the lower humid layer through the moist enthalpy conservation, and gives a sink (source)
in the mass conservation equation in the lower (upper) layer. The coefficient β is determined
by the background stratification in the parent primitive-equations model, see Lambaerts et
al. (2011a). Due to the convective mass exchange the same flux leads to appearance of the
Rayleigh drag in the upper-layer momentum equation. At the same time, condensation gives
a sink in the bulk moisture content Q1 in the lower layer. If surface evaporation E is present,
it provides a source of moisture in the lower layer. Finally, condensation and moisture content
are related by a simple relaxation relation:
Q1 − Qs
H(Q1 − Qs ),
(2)
τ
where H(.) is the Heaviside (step) function, and Qs is a saturation value. This scheme is of the
type used in general circulation models (Betts and Miller 1986). In what follows the simplest
parameterisation with a constant Qs is used. Note that the condensed water is dropped off
the dynamics, and we do not consider the inverse phase transition liquid water - water vapor
in this simplified model. (Hence the evaporation E, if any, is the surface one in the model,
and in this sense there is no difference between condensation and precipitation, and they will
P1 =